Some thoughts on Ando's theorem and Parrott's example

Abstract

In this short note, we present an elementary proof of Ando's theorem within a restricted class P of homomorphisms modeled after Parrott's example. We also show by explicit estimation that the cb-norm of the contractive homomorphism ρ of the tri-disc algebra, induced by the commuting triple of Parrott, exceeds 1. Indeed, we construct a polynomial P with matrix coefficients with the property ||ρ(P)||>||P||∞. In particular, we show that there are contractive homomorphisms of the tri-disc algebra which are not even 2-contractive.